Q. 51 Use the result of Problem 50 and... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 8: Q. 51 (page 543)

Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show that
$\cot \frac{C}{2} = \frac{s - c}{r}$ where $s = \frac{1}{2} (a + b + c)$ .

Short Answer

Expert verified

It is proved that $\cot \frac{C}{2} = \frac{s - c}{r}$ .

Step by step solution

Step 1. Given Informtion

Given that $s = \frac{1}{2} (a + b + c)$

use localid="1650175628630" $r = \frac{c \sin \frac{A}{2} \sin \frac{B}{2}}{\cos \frac{C}{2}}$

localid="1650175390705" $\sin \frac{C}{2} = \sqrt{\frac{(s - a) (s - b)}{a b}} \cot = \frac{\sin}{\cos}$

Step 2. Proving the statement

$r = \frac{c \sin \frac{A}{2} \sin \frac{B}{2}}{\cos \frac{C}{2}} \cos \frac{C}{2} = \frac{c \sin \frac{A}{2} \sin \frac{B}{2}}{r}$

$\frac{\cos \frac{C}{2}}{\sin \frac{C}{2}} = \frac{c}{r} \frac{\sin \frac{A}{2} \sin \frac{B}{2}}{\sin \frac{C}{2}} \cot \frac{C}{2} = \frac{c}{r} \frac{\sin \frac{A}{2} \sin \frac{B}{2}}{\sin \frac{C}{2}}$

Now.

$\cot \frac{C}{2} = \frac{c}{r} \frac{\sqrt{\frac{(s - b) (s - c)}{b c}} \sqrt{\frac{(s - a) (s - c)}{a c}}}{\sqrt{\frac{(s - a) (s - b)}{a b}}} \cot \frac{C}{2} = \frac{c}{r} \sqrt{\frac{a b (s - a) (s - b) {(s - c)}^{2}}{a b c^{2} (s - a) (s - b)}} \cot \frac{C}{2} = \frac{c}{r} \sqrt{\frac{{(s - c)}^{2}}{c^{2}}} \cot \frac{C}{2} = \frac{c}{r} \frac{s - c}{c} = \frac{s - c}{r}$

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91影视

Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show that
$\cot \frac{C}{2} = \frac{s - c}{r}$ where $s = \frac{1}{2} (a + b + c)$ .

Short Answer

Step by step solution

Step 1. Given Informtion

Step 2. Proving the statement

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91影视

Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show thatcotC2=s-crwheres=12a+b+c.

Short Answer

Step by step solution

Step 1. Given Informtion

Step 2. Proving the statement

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Discrete Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use the result of Problem 50 and the results of Problems 56 and 57 in Section 8.3 to show that
$\cot \frac{C}{2} = \frac{s - c}{r}$ where $s = \frac{1}{2} (a + b + c)$ .